Gramacy-Lee's Function No.01

I. Mathematical Expression:

$$f(x)=\frac{\displaystyle \sin\left(10\pi x\right)}{\displaystyle 2x}+\left(x-1\right)^4$$

where:

\(\bullet\) \(0.5 \leq x \leq 2.5\)

\(\bullet\) \(f_{min}(x^*)=-0.869011134989500\)

\(\bullet\) \(x^*=0.548563444114526\) (determined by us using MapleSoft 2015)

II. Citation Policy:

If you publish material based on databases obtained from this repository, then, in your acknowledgments, please note the assistance you received by using this repository. This will help others to obtain the same data sets and replicate your experiments. We suggest the following pseudo-APA reference format for referring to this repository:

Ali R. Al-Roomi (2015). Unconstrained Single-Objective Benchmark Functions Repository [https://www.al-roomi.org/benchmarks/unconstrained]. Halifax, Nova Scotia, Canada: Dalhousie University, Electrical and Computer Engineering.

Here is a BiBTeX citation as well:

@MISC{Al-Roomi2015,
author = {Ali R. Al-Roomi},
title = {{Unconstrained Single-Objective Benchmark Functions Repository}},
year = {2015},
address = {Halifax, Nova Scotia, Canada},
institution = {Dalhousie University, Electrical and Computer Engineering},
url = {https://www.al-roomi.org/benchmarks/unconstrained}
}

III. 2D-Plot:

IV. MATLAB M-File:

% Gramacy-Lee's Function # 1
% Range of initial points: 0.5 <= x <= 2.5
% Global minima: x=0.548563444114526
% f(x)=-0.869011134989500
% Coded by: Ali R. Alroomi | Last Update: 23 July 2015 | www.al-roomi.org
 
clear
clc
warning off
 
xmin=0.5;
xmax=2.5;
R=100000; % steps resolution
x=xmin:(xmax-xmin)/R:xmax;
 
for i=1:length(x)
    f(i)=sin(10*pi*x(i))/(2*x(i))+(x(i)-1)^4;
end
 
plot(x,f,'r','LineWidth',2);grid;set(gca,'FontSize',12);
xlabel('x','FontName','Times','FontSize',20,'FontAngle','italic');
ylabel('f(x)','FontName','Times','FontSize',20,'FontAngle','italic');

Click here to download m-file

V. References:

[1] Robert B. Gramacy, and Herbert K.H. Lee, "Cases for the Nugget in Modeling Computer Experiments," Statistics and Computing, vol. 22, no. 3, pp. 713-722, May 2012.
[2] Ali R. Alroomi, "The Farm of Unconstrained Benchmark Functions," University of Bahrain, Electrical and Electronics Department, Bahrain, Oct. 2013. [Online]. Available: http://www.al-roomi.org/cv/publications